Perfect State Transfer in Weighted Cubelike Graphs

Abstract

A continuous-time quantum random walk describes the motion of a quantum mechanical particle on an underlying graph. The graph itself is associated with a Hilbert space of dimension equal to the number of vertices. The dynamics of the walk is governed by the unitary operator U(t) = e iAt , where A is the adjacency matrix of the graph. An important notion in the quantum random walk is the transfer of a quantum state from one vertex to another. If the fidelity of the transfer is unity, we call it a perfect state transfer. Many graph families have been shown to admit PST or periodicity, including cubelike graphs. These graphs are unweighted. In this paper, we generalize the PST or periodicity of cubelike graphs to that of weighted cubelike graphs. We characterize the weights for which they admit PST or show periodicity, both at time t = π 2 .

Quantum simulation of perfect state transfer on weighted cubelike graphs

“…We have used Hamiltonian simulation techniques to construct efficient circuits for continuous-time quantum random walks. We have verified the theoretical results of [5] and [15] that PST or periodicity on integral weighted cubelike graphs occurs at time t = π 2 , where weights are determined by Theorem 2.4. In the future, we plan to construct efficient quantum circuits for quantum walks on weighted abelian Cayley graphs.…”

“…Theorem 2.4. [5,15] Let f : Z n 2 → Z be an integer-valued function. For x ∈ Z n 2 , define a subset O x = {y ∈ Z n 2 : x|y mod 2 = 1}.…”

“…Note. Although PST or periodicity in weighted cubelike graph mentioned in [15] was done independently, it was only later that the authors realized that its generalized version, viz., PST on weighted abelian Cayley graph, has already been proved in another paper [5].…”

“…We use this fact to construct an efficient quantum circuit for the quantum walk on cubelike graphs. In [5,15], a characterization of integer weighted cubelike graphs is given that exhibit periodicity or PST at time t = π/2. We use our circuits to demonstrate PST or periodicity in these graphs on IBM's quantum computing platform [1,10].…”